Algorithms and Software for Large-Scale Simulation of Reactive Systems _______________________________ Ananth Grama Coordinated Systems Lab Purdue University

Molecular Simulation Methods Ab-initio methods (few approximations but slow) DFT CPMD Electron and nuclei treated explicitly Classical atomistic methods (more approximations) Classical molecular dynamics Monte Carlo Brownian dynamics No electronic degrees of freedom. Electrons are approximated through fixed partial charges on atoms. Continuum methods (no atomistic details)

Statistical and continuum methods ps ns ss ms nm mm mm Ab-initio methods Atomistic methods

V = V bond + V angle + V dihedral + V LJ + V Elecrostatics Simplified Interactions in Classical MD Simulations

Implementation of Classical Interactions Molecular topologies are fixed, so bonded interactions are implemented as static neighbor lists Non-bonded interactions are implemented as dynamic neighbor lists Usually not updated at every time step Only two body interactions, so relatively easy to implement.

Reactive systems Chemical reactions correspond to association and dissociation of chemical bonds Classical simulations cannot simulate reactions ab-initio methods calculate overlap of electron orbitals to model chemical reactions ReaX force field postulates a classical bond order interaction to mimic the association and dissociation of chemical bonds 1 1 van Duin et al, J. Phys. Chem. A, 105, 9396 (2001)

Bond order interaction 1 van Duin et al, J. Phys. Chem. A, 105, 9396 (2001) Bond order for C-C bond Uncorrected bond order: Where  is for  and  bonds  The total uncorrected bond order is sum of three types of bonds Bond order requires correction to account for the correct valency

Upon correction, the bond order between a pair of atoms depends on the uncorrected bond orders of the neighbors of each atoms The bond orders rapidly decay to zero as a function of distance so it is reasonable to construct a neighbor list for efficient computation of bond orders Bond Order Interaction

Neighbor Lists for Bond Order Efficient implementation critical for performance Implementation based on an oct-tree decomposition of the domain For each particle, we traverse down to neighboring octs and collect neighboring atoms Has implications for parallelism (issues identical to parallelizing multipole methods)

Bond Order : Choline

Bond Order : Benzene

Other Local Energy Terms Other interaction terms common to classical simulations, e.g., bond energy, valence angle and torsion, are appropriately modified and contribute to non-zero bond order pairs of atoms These terms also become many body interactions as bond order itself depends on the neighbors and neighbor’s neighbors Due to variable bond structure there are other interaction terms, such as over/under coordination energy, lone pair interaction, 3 and 4 body conjugation, and three body penalty energy

Non Bonded van der Waals Interaction The van der Waals interactions are modeled using distance corrected Morse potential Where R(r ij ) is the shielded distance given by

Electrostatics Shielded electrostatic interaction is used to account for orbital overlap of electrons at closer distances Long range electrostatics interactions are handled using the Fast Multipole Method (FMM).

Charge Equilibration (QEq) Method The fixed partial charge model used in classical simulations is inadequate for reacting systems. One must compute the partial charges on atoms at each time step using an ab-initio method. We compute the partial charges on atoms at each time step using a simplified approach call the Qeq method.

Charge Equilibration (QEq) Method Expand electrostatic energy as a Taylor series in charge around neutral charge. Identify the term linear in charge as electronegativity of the atom and the quadratic term as electrostatic potential and self energy. Using these, solve for self-term of partial derivative of electrostatic energy.

Qeq Method We need to minimize: subject to: where

Qeq Method

From charge neutrality, we get:

Qeq Method Let where or

Qeq Method Substituting back, we get: We need to solve 2n equations with kernel H for s i and t i.

Qeq Method Observations: –H is dense. –The diagonal term is J i –The shielding term is short-range –Long range behavior of the kernel is 1/r

Implementation, Performance, and Validation

Serial Performance: Scaling

Parallel Performance Reactive and non-reactive MD simulations on 131K BG/L processors. Total execution time per MD step as a function of the number of atoms for 3 algorithms: QMMD, ReaxFF,conventional MD [Goddard, Vashistha, Grama]

Parallel Performance Total execution (circles) and communication (squares) times per MD time for the ReaxFF MD with scaled workloads—36,288 x p atom RDX systems (p = 1,..,1920).

Current Development Efforts Development and validation of parallel version of next generation Reax code. Integration into LAMMPS.

Planned Development Efforts Interface with conventional MD Interface with continuum models Validation in the context of surface contact for RF MEMS device